What is the integral of cos(x)(4+8sin^2(x)) ?

The integral of cos(x)(4+8sin^2(x)) is 4sin(x)+(8sin^3(x))/3+C

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integral of cos(x)(4+8sin^2(x))

\int cos (x) (4 + 8 sin^{2} (x)) d x

4 sin (x) + \frac{8 sin ^{3} ( x )}{3} + C

Solution steps

\int cos (x) (4 + 8 sin^{2} (x)) d x

Apply u-substitution

: \int 4 + 8 u^{2} d u

= \int 4 + 8 u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 4 d u + \int 8 u^{2} d u

\int 4 d u = 4 u

\int 8 u^{2} d u = \frac{8 u ^{3}}{3}

= 4 u + \frac{8 u ^{3}}{3}

Substitute back

u = sin (x)

= 4 sin (x) + \frac{8 sin ^{3} ( x )}{3}

Add a constant to the solution

= 4 sin (x) + \frac{8 sin ^{3} ( x )}{3} + C

x \to 0 lim (x^{2} cos (\frac{4}{x}))

x^{\frac{1}{- 3}}

y = ln (x^{2} - 2)

\frac{d}{d x} (6 sin^{6} (x))

What is the integral of cos(x)(4+8sin^2(x)) ?
The integral of cos(x)(4+8sin^2(x)) is 4sin(x)+(8sin^3(x))/3+C